INVINCIBLE ORIGINALITY OF SRINIVASA RAMANUJAN

1. Context

2. Srinivasa Ramanujan Early life

• Srinivasa Ramanujan (1887-1920) was an Indian mathematician who, with almost no formal training in mathematics, made extraordinary contributions to number theory, mathematical analysis, infinite series, and continued fractions. He had no traditional degree or education but significantly contributed to mathematics. His mathematical equations are hard to solve. No one discovers his derivations properly, and they got only the formulas or some distorted number of equations.
• Ramanujan had a brief life span of only 32 years. He died of TB at the age of 32; but left a collection of papers that contained more than 3,000 formulas and identities, many of which are yet to be proved by other mathematicians. Many of the formulas in this collection were so complex that they had been rejected by many eminent mathematicians of his time, including Hardy and Littlewood.
• Srinivasa Ramanujan's work has found application in physics and engineering. He developed rapidly as a mathematician under the guidance of G.H. Hardy, who brought him to Cambridge University in 1916. He published more than 340 mathematical papers spanning various mathematical areas.
• Ramanujan is now recognized as one of the greatest Indian mathematicians ever and one of the most influential mathematicians. He was selected as a Fellow of the Royal Society in 1914 at the unusually young age of 26.

3. Ramanujan Machine

• In early 2021, a team of Israeli scientists announced a software tool called the Ramanujan Machine that creates mathematical conjectures and equations without proof. Mathematicians then prove or disprove these conjectures, thereby establishing theorems.
• Conjectures in mathematics shed light on newer frontiers that otherwise lurk in tenebrous
• concerns. Srinivasa Ramanujan was famous for such conjectures. From 1904 till his passing in 1920, Ramanujan recorded more than 3,000 equations that were mostly conjectured because he did not supply proof.

4. Contributions of Ramanujan

• Ramanujan's mathematical discoveries were made independently from any formal training in pure mathematics, and he even made unproven conjectures at the time. while some of his results were proven by others, many remain true but as yet unproven. He has also given several mathematical identities, such as the Reimann series, elliptical integrals, hypergeometric series, and Zeta function.
• Ramanujan memorized formulas and their values for many special functions (such as elliptic integrals) that were not available in texts. He developed new analytical capabilities, which allowed him to compute complicated values of these functions and generalizations of some functions like the gamma function, zeta function, and beta function, which came to be known as Ramanujan's formulas.
• He discovered a long list of new ideas to solve many challenging mathematical problems, which gave a significant impetus to the development of game theory. His contribution to game theory is purely based on intuition and natural talent and remains unrivaled to this day.
• 1729 is known as the Ramanujan number. It is the smallest number that can be expressed as the sum of two different cubes in two different ways. 1729 is the sum of the cubes of 10 and 9-cube of 10 is 1000 and the cube of 9 is 729 adding the two numbers results in 1729. 1729 is also the sum of the cubes of 12 and 1, cubes of 12 is 1728, and cube of 1 is 1 adding the two results is 1729.

5. Final breakthrough

• Srinivasa Ramanujan did not keep all his discoveries to himself but continued to send his works to International mathematicians.
• In 1912, he was appointed to the position of clerk in the Madras Post Trust Office, where the manager, S.N. Aiyar encouraged him to reach out to G.H. Hardy, a famous mathematician at Cambridge University.
• In 1913, he sent a famous letter to Hardy, in which he had attached 120 theorems as a sample of his work.
• Hardy along with another mathematician at Cambridge, J.E.Littlewood analyzed his work and concluded it to be a work of true genius.
• After this, his journey and recognition as one of the greatest mathematicians started.

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